# double factorial

The double factorial of a positive integer $n$ is the product $n!!$ of the positive integers less than or equal to $n$ that have the same parity as $n$, that is,

 $n!!=n(n-2)(n-4)\cdots k_{n}$

where $k_{n}$ denotes $1$ if $n$ is an odd number and $2$ if $n$ is an even number.

For example,

 $7!!=7\cdot 5\cdot 3\cdot 1=105$
 $10!!=10\cdot 8\cdot 6\cdot 4\cdot 2=3840$

Note that $n!!$ is not the same as $(n!)!$.

Observe that $(2n)!!=2^{n}n!$ and $(2n+1)!!=\frac{(2n+1)!}{2^{n}n!}$.

Title double factorial DoubleFactorial 2013-03-22 12:24:54 2013-03-22 12:24:54 drini (3) drini (3) 9 drini (3) Definition msc 05A10